Laws of large numbers for cooperative St. Petersburg gamblers
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LAWS OF LARGE NUMBERS FOR COOPERATIVE ST. PETERSBURG GAMBLERS ´ndor Cso ¨ rgo ˝ (Szeged) and Gordon Simons (Chapel Hill) Sa Dedicated to Endre Cs´ aki and P´ al R´ev´esz for their seventieth birthdays
Abstract General linear combinations of independent winnings in generalized St. Petersburg games are interpreted as individual gains that result from pooling strategies of different cooperative players. A weak law of large numbers is proved for all such combinations, along with some almost sure results for the smallest and largest accumulation points, and a considerable body of earlier literature is fitted into this cooperative framework. Corresponding weak laws are also established, both conditionally and unconditionally, for random pooling strategies.
1. Introduction Let X be Paul’s gain, in ducats, in a St. Petersburg game played with a possibly biased coin, for which the probability of ‘heads’ is p ∈ (0, 1), the game designed so that P{X = rj } = q j−1 p, j ∈ N := {1, 2, . . .}, where q = 1 − p and r = 1/q = 1/(1 − p). Then, setting y = max{j ∈ Z : j ≤ y}, y = min{j ∈ Z : j ≥ y}, for the two kinds of integer parts, and y = y − y = y + −y for the fractional part of y ∈ R, where Z = {0, ±1, ±2, . . .} and R is the set of real numbers, and letting logr stand for the logarithm to base r, we see that the distribution function of X is if x < r, 0, log r x Fp (x) = P{X ≤ x} = (1) 1 − q logr x = 1 − r , if x ≥ r. x Mathematics subject classification number: 60F05. Key words and phrases: St. Petersburg games, cooperative players, deterministic and random pooling strategies, entropy, laws of large numbers. Read by the first author at the International Conference on Probability and Statistics “Endre Cs´ aki and P´ al R´ ev´ esz are 70”, June 17–19, 2004, Budapest; partially supported by the Hungarian National Foundation for Scientific Research, Grants T–034121 and T–048360. 0031-5303/2005/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
¨ rgo ˝ and g. simons s. cso
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∞ ∞ Since E(X) = p j=1 rj q j−1 = p j=1 r = ∞, Peter, the banker, would ask for any large sum for the gamble, while Paul would not risk even a moderate amount of x ducats because P{X > x} = rlogr x /x is small. Thus Nicolaus Bernoulli’s 292-yearold question (traditionally asked for an unbiased coin with p = 1/2), demanding Paul’s ‘fair price’ for the game, led to the St. Petersburg paradox; see for example [16] for some historical references. In what follows we refer to the extension above as St. Petersburg(p), the St. Petersburg game, situation, distribution or random variable with parameter p ∈ (0, 1). Let X1 , X2 , . . . be Paul’s gains in a sequence of independent repetitions of the St. Petersburg(p) game, given on a probability space (Ω, A, P), so that Paul’s total winnings in n games is X1 + · · · + Xn . Then n k=1 Xk P p (2) −→ , n logr n q P
where −→ denotes convergence in probability as n → ∞, and, with a.s. abbreviating “almost surely” or “almost sure” as the case dictates, n n X
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